Calculus Examples

$y = x^{3} - 3 x^{2} - 2$ , $(1, - 4)$

Step 1

Find the first derivative and evaluate at $x = 1$ and $y = - 4$ to find the slope of the tangent line.

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Step 1.1

Differentiate.

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Step 1.1.1

By the Sum Rule, the derivative of $x^{3} - 3 x^{2} - 2$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 3 x^{2}] + \frac{d}{d x} [- 2]$ .

$\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 3 x^{2}] + \frac{d}{d x} [- 2]$

Step 1.1.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$3 x^{2} + \frac{d}{d x} [- 3 x^{2}] + \frac{d}{d x} [- 2]$

Step 1.2

Evaluate $\frac{d}{d x} [- 3 x^{2}]$ .

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Step 1.2.1

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3 x^{2}$ with respect to $x$ is $- 3 \frac{d}{d x} [x^{2}]$ .

$3 x^{2} - 3 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 2]$

Step 1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$3 x^{2} - 3 (2 x) + \frac{d}{d x} [- 2]$

Step 1.2.3

Multiply $2$ by $- 3$ .

$3 x^{2} - 6 x + \frac{d}{d x} [- 2]$

Step 1.3

Differentiate using the Constant Rule.

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Step 1.3.1

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

$3 x^{2} - 6 x + 0$

Step 1.3.2

Add $3 x^{2} - 6 x$ and $0$ .

$3 x^{2} - 6 x$

Step 1.4

Evaluate the derivative at $x = 1$ .

$3 {(1)}^{2} - 6 \cdot 1$

Step 1.5

Simplify.

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Step 1.5.1

Simplify each term.

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Step 1.5.1.1

One to any power is one.

$3 \cdot 1 - 6 \cdot 1$

Step 1.5.1.2

Multiply $3$ by $1$ .

$3 - 6 \cdot 1$

Step 1.5.1.3

Multiply $- 6$ by $1$ .

$3 - 6$

Step 1.5.2

Subtract $6$ from $3$ .

$- 3$

Step 2

Plug the slope and point values into the point-slope formula and solve for $y$ .

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Step 2.1

Use the slope $- 3$ and a given point $(1, - 4)$ to substitute for $x_{1}$ and $y_{1}$ in the point-slope form $y - y_{1} = m (x - x_{1})$ , which is derived from the slope equation $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (- 4) = - 3 \cdot (x - (1))$

Step 2.2

Simplify the equation and keep it in point-slope form.

$y + 4 = - 3 \cdot (x - 1)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $- 3 \cdot (x - 1)$ .

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Step 2.3.1.1

Rewrite.

$y + 4 = 0 + 0 - 3 \cdot (x - 1)$

Step 2.3.1.2

Simplify by adding zeros.

$y + 4 = - 3 \cdot (x - 1)$

Step 2.3.1.3

Apply the distributive property.

$y + 4 = - 3 x - 3 \cdot - 1$

Step 2.3.1.4

Multiply $- 3$ by $- 1$ .

$y + 4 = - 3 x + 3$

Step 2.3.2

Move all terms not containing $y$ to the right side of the equation.

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Step 2.3.2.1

Subtract $4$ from both sides of the equation.

$y = - 3 x + 3 - 4$

Step 2.3.2.2

Subtract $4$ from $3$ .

$y = - 3 x - 1$

Step 3